column results in: ,which is an exellent approximation considering that we used only 10 samples from relatively low values of , and taking the average of the column results in: , which is very close to the most precisely determined value of the fine structure constant to date,and matches the latest Codata value perfectly!

So, in theory, if we had sufficiently large values of

, say , to about or so...then we can simply take the average of sufficiently many random samples of to get to as many decimal places as we like, and thereby generate the entire sequence of primes in sequential order!

It's essentially the same principle as flipping a coin sufficiently many times and averaging out the results in order to get as close to

as we like.

I really like the idea of using one erratic sequence to generate another. It's kind of like fighting fire with fire.

Re: Polygonal Number Counting Function

Hi Don Blazys;

Welcome to the forum. I am familiar with some of your ideas and the controversy that surrounds them. I do not say that you are the cause of the arguing and ad hominem attacks that follow your work on other forums and blogs. If it follows you here then I must say I will moderate it strongly.

Name calling or personal attacks regardless of the reputation of the aggressor will be deleted immediately.

In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Re: Polygonal Number Counting Function

___________________________________________Math is mysterious.______________________________________________

Everyone loves a good mystery, and math is not only one of the most important tools that scientists use in solving the riddles and mysteries of the universe, but it is also a fascinating subject in its own right, and contains some of the most perplexing puzzles and profound problems known to mankind.

The counting function in post #1 is an exellent example of just how mysterious some math problems can be.

How many polygonal numbers of order greater than 2 are there less than or equal to

? Nobody knows! The above counting function can be used to approximate the answer, but the exact value of remains a mystery.

Why does approximating the number of polygonal numbers of order greater than 2 to a high degree of accuracyrequire the "running" of the fine structure constant which is by far the most important constant in all of physics? Again, nobody knows! Google searching the phrase " reflexive polygons in string theory" brings up all kinds of results showing that polygonal numbers are at the very core of string theory, but so far, that entire issue remains a mystery!

__________________________________________Math is challenging._____________________________________________

Everybody loves a challenge. Indeed, people have climbed Mt. Everest and swam across the English Channel simplybecause it was a challenge and for no reason other than "it was there". A life without challenges is dull, boring and hardlyworth living while a life that is filled with challenges is extraordinarily interesting and (most importantly), loads of fun!

The counting function in post #1 is a perfect example of just how challenging some math problems can be.

Seperating the polygonal numbers of order greater than 2 from the rest of the polygonal numbers is analogous to seperating the composite numbers from the prime numbers. Both are extraordinarily hard to do, and doing either results in sequences that are absolutely random and erratic, yet follow certain other laws in a manner that is quite predictable.

Polygonal numbers of order greater than 2 have only been counted up to

. That's the current "world record". A lot of coders tried very hard to break that record, but most of them gave up after their computers either crashed or ground to a halt. However, I'm sure that other coders will continue trying to break that record, not only because breaking records is a fun and challenging thing to do, but because the counting function in post #1 is perhaps the most unique counting function in all of mathematics, and as such, gets first page ranking by Google and is even referenced in the Online Encyclopedia of Integer Sequences. It is certainly the only counting function that involves polygonal numbers.

I put it here, just in case you might want to try and break that record.

If you don't, then please lock this thread and I will continue having fun elsewhere.

That should be interesting, I thought, given that some mathematicians think it cannot be done. But I'll keep an open mind. After all, I think the aquatic ape theory is correct in the face of most scientific thinking and that humans have more than 5 senses despite what they tell you in biology text books, so why not try out this idea too.

Now I'm uncertain exactly what your generator is. Obviously, my brain is only splashing about in the wake of yours (and I'm serious, not trying to be rude I promise ) but I had a problem with this.

Re: Polygonal Number Counting Function

I'm very interested in why it works and I don't see why that requires any computing power at all.

Thanks Bob.

It works only "in theory". Actually proving that it works may or may not be possible.

So far, all I have managed to demonstrate is that the general form of the counting function is probably correct. However, my notebooks contain dozens of variations on that form, all of which are highly accurate to

, and the only way to determine which of those variations will remain highly accurate is to determine higher values of .

Quoting bob bundy:

...some mathematicians think it cannot be done.

Google searching "prime number generating formulas" shows that there are many such formulas,some of which are quite clever and interesting. The problem is that none of them are efficient enough to be of any practical value.

Now, generating primes by counting polygonal numbers of order greater than 2 may or may not turn out to be practical,but again, the only way to actually test the efficiency of this method is to determine larger values of

.

Since this is the only known method which generates all the primes and only the primes in sequential order, I think that testing its efficiency would be interesting, informative, and a lot of fun.

Re: Polygonal Number Counting Function

Note that the "previous prime" is simply the floor function of the "previous value".

Quoting bob bundy:

What I was hoping for is a proof...

Carl Gauss discovered the "simple" prime number counting function:

while still in his teens. Proving that it works all the way into

requiredanother century of hard labor by some of the worlds greatest mathematicians.

Now, my counting function for polygonal numbers of order greater than 2is very, very sophisticated in that it involves not only the above prime numbercounting function, but

and as well!

Thus, proving its convergence with

would be extraordinarily difficult,and proving that it generates all of the primes and only the primes in sequential order would actually require a proof of the Riemann hypothesis, which may or may not be provable!

Quoting bob bundy:

Trouble with this seems to me to be that this search will never end.

From my point of view,

.Thus, if the search will never end, then the fun will never end, and that's a good thing!